Krylov complexity for Lin-Maldacena geometries and their holographic duals

We compute the rate of growth of operator size in matrix models by probing the Lin-Maldacena class of geometries with classical probes. We consider massive point particle probes whose proper momentum equals the size of the gauge invariant operator in the matrix model. We work out the example of the BMN Plane Wave Matrix Model using the electrostatic approach and the method of background fluxes. We also work out complexities in the D2 brane as well as NS5 brane limits of the BMN matrix model along with an example of the irrelevant deformation namely the non-Abelian T-dual of $AdS_5 \times S^5$. Finally, we carry out a possible calculation of the Krylov complexity on the matrix model counterpart by using a simple reduction ansatz known as the pulsating fuzzy sphere model. We outline an algorithm to define Krylov basis elements for the matrix model and compute a few Lanczos coefficients. Our analysis reveals that both the Krylov basis states as well as Lanczos coefficients are uniquely fixed in terms of the mass parameter of the matrix model.